Is gabapentin a barbiturate or benzodiazepine?

Neither. It is an anti-seizure med. However, it may have some effects on the gaba receptor, though this may not be it's main mechanism of action. It can be sedating, like the other 2 classes of drugs.
Neither. Gabapentin works on voltage gated calcium channels to decrease release of excitatory neurotransmitters. Barbs/benzos work on the gaba receptor increasing its inhibitory function. Interestingly, Gabapentin was designed to work on gaba but in fact has nothing to do with gaba.